Simplify (((3xy^-5)^3)/((x^-2y^2)^-4))

$(\frac{{(3 x y^{- 5})}^{3}}{{(x^{- 2} y^{2})}^{- 4}})$

Step 1

Remove parentheses.

$\frac{{(3 x y^{- 5})}^{3}}{{(x^{- 2} y^{2})}^{- 4}}$

Step 2

Move

${(x^{- 2} y^{2})}^{- 4}$ to the numerator using the negative exponent rule

$\frac{1}{b^{- n}} = b^{n}$ .

${(3 x y^{- 5})}^{3} {(x^{- 2} y^{2})}^{4}$

Step 3

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

${(3 x \frac{1}{y^{5}})}^{3} {(x^{- 2} y^{2})}^{4}$

Step 4

Multiply

$3 x \frac{1}{y^{5}}$ .

Combine

$3$ and

$\frac{1}{y^{5}}$ .

${(x \frac{3}{y^{5}})}^{3} {(x^{- 2} y^{2})}^{4}$

Combine

$x$ and

$\frac{3}{y^{5}}$ .

${(\frac{x \cdot 3}{y^{5}})}^{3} {(x^{- 2} y^{2})}^{4}$

Step 5

Move

$3$ to the left of

$x$ .

${(\frac{3 \cdot x}{y^{5}})}^{3} {(x^{- 2} y^{2})}^{4}$

Step 6

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to

$\frac{3 x}{y^{5}}$ .

$\frac{{(3 x)}^{3}}{{(y^{5})}^{3}} {(x^{- 2} y^{2})}^{4}$

Apply the product rule to

$3 x$ .

$\frac{3^{3} x^{3}}{{(y^{5})}^{3}} {(x^{- 2} y^{2})}^{4}$

Step 7

Raise

$3$ to the power of

$3$ .

$\frac{27 x^{3}}{{(y^{5})}^{3}} {(x^{- 2} y^{2})}^{4}$

Step 8

Multiply the exponents in

${(y^{5})}^{3}$ .

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{27 x^{3}}{y^{5 \cdot 3}} {(x^{- 2} y^{2})}^{4}$

Multiply

$5$ by

$3$ .

$\frac{27 x^{3}}{y^{15}} {(x^{- 2} y^{2})}^{4}$

Step 9

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{27 x^{3}}{y^{15}} {(\frac{1}{x^{2}} y^{2})}^{4}$

Step 10

Combine

$\frac{1}{x^{2}}$ and

$y^{2}$ .

$\frac{27 x^{3}}{y^{15}} {(\frac{y^{2}}{x^{2}})}^{4}$

Step 11

Apply the product rule to

$\frac{y^{2}}{x^{2}}$ .

$\frac{27 x^{3}}{y^{15}} \cdot \frac{{(y^{2})}^{4}}{{(x^{2})}^{4}}$

Step 12

Combine.

$\frac{27 x^{3} {(y^{2})}^{4}}{y^{15} {(x^{2})}^{4}}$

Step 13

Apply basic rules of exponents.

Multiply the exponents in

${(y^{2})}^{4}$ .

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{27 x^{3} y^{2 \cdot 4}}{y^{15} {(x^{2})}^{4}}$

Multiply

$2$ by

$4$ .

$\frac{27 x^{3} y^{8}}{y^{15} {(x^{2})}^{4}}$

Multiply the exponents in

${(x^{2})}^{4}$ .

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{27 x^{3} y^{8}}{y^{15} x^{2 \cdot 4}}$

Multiply

$2$ by

$4$ .

$\frac{27 x^{3} y^{8}}{y^{15} x^{8}}$

Step 14

Cancel the common factor of

$x^{3}$ and

$x^{8}$ .

Factor

$x^{3}$ out of

$27 x^{3} y^{8}$ .

$\frac{x^{3} (27 y^{8})}{y^{15} x^{8}}$

Cancel the common factors.

Factor

$x^{3}$ out of

$y^{15} x^{8}$ .

$\frac{x^{3} (27 y^{8})}{x^{3} (y^{15} x^{5})}$

Cancel the common factor.

$\frac{x^{3} (27 y^{8})}{x^{3} (y^{15} x^{5})}$

Rewrite the expression.

$\frac{27 y^{8}}{y^{15} x^{5}}$

Step 15

Cancel the common factor of

$y^{8}$ and

$y^{15}$ .

Factor

$y^{8}$ out of

$27 y^{8}$ .

$\frac{y^{8} \cdot 27}{y^{15} x^{5}}$

Cancel the common factors.

Factor

$y^{8}$ out of

$y^{15} x^{5}$ .

$\frac{y^{8} \cdot 27}{y^{8} (y^{7} x^{5})}$

Cancel the common factor.

$\frac{y^{8} \cdot 27}{y^{8} (y^{7} x^{5})}$

Rewrite the expression.

$\frac{27}{y^{7} x^{5}}$

Simplify (((3xy^-5)^3)/((x^-2y^2)^-4))

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